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課程名稱 |
代數一
Algebra (Ⅰ) |
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開課學期 |
100-1 |
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授課對象 |
理學院 數學研究所 |
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授課教師 |
于 靖 |
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課號 |
MATH7105 |
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課程識別碼 |
221 U3830 |
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班次 |
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學分 |
3 |
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全/半年 |
半年 |
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必/選修 |
選修 |
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上課時間 |
星期一3,4(10:20~12:10)星期四7,8(14:20~16:20) |
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上課地點 |
天數305天數305 |
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備註 |
研究所數學組基礎課。
總人數上限：30人
外系人數限制：5人 |
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Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1001algebra1 |
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課程簡介影片 |
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核心能力關聯 |
核心能力與課程規劃關聯圖 |
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課程大綱
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為確保您我的權利,請尊重智慧財產權及不得非法影印
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課程概述 |
1. Free groups, Simple groups, Profinite groups, Group Representations, Group Extensions, Topological groups.
2. Finitely generated modules over PID.
3. UFD, Localizations, Projective Modules, Noetherian rings. Dedekind domains.
4. Infinite Galois theory.
5. Categories, Functors,Tensor Products.
6. Division algebras, Semisimple algebras. |
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課程目標 |
Course Goal：
For students who are interested in various related fields, to equip them with a solid background in algebra.
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課程要求 |
建議先修:Undergraduate algebra, including Sylow theorems and group actions, Galois theory.
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預期每週課後學習時數 |
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Office Hours |
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指定閱讀 |
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參考書目 |
N. Jacobson: Basic Algebra II, 2010
N. Jacobson: Basic Algebra I, 2009
S. Lang, Algebra, GTM Springer, revised 3rd ed. 2002.
M. Artin, Algebra, 2nd ed. 2011, Prentice Hall.
A. Cox, J. Little, and O'shea: Ideals, Varieties, and Algorithms, 2nd. ed. Springer. |
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評量方式
(僅供參考) |
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No. |
項目 |
百分比 |
說明 |
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1. |
Oral presentations |
25% |
Every week, there will be one hour for students to present their homeworks and individual studies. |
2. |
Final exam |
25% |
6 hours written exam in one day. |
3. |
Homeworks |
25% |
To be graded by the assistant |
4. |
Mid-term Exam |
25% |
Taking home open book examination. |
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週次 |
日期 |
單元主題 |
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第1週 |
9/19,9/15 |
Review group theory, families of groups, solvable groups, group extensions, review PID, UFD, localization, modules. |
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第2週 |
9/26,9/22 |
Finitely generated modules over PID, Jordan-Holder theorem, Categories, Functors, Universal objects, Simple modules, Completely reducible modules, Semi-Simplifications. |
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第3週 |
10/03,9/29 |
Modules of finite length, Noetherian modules, Artinian modules, Categories, Functors. |
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第4週 |
10/10,10/06 |
Integral extensions, Free groups, Free products, Duality theory for finite abelian groups, Tensor product. |
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第5週 |
10/17,10/13 |
Products. Coproducts. Npetherian rings, Artinian rings, Associated primes. |
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第6週 |
10/24,10/20 |
Norm, Trace, Finite Free Resolutions. Symmetric powers. Exterior Powers, polynomial rings. |
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第7週 |
10/31,10/27 |
Direct limits, Inverse limits, profinite groups, group rings, group algebras. |
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第8週 |
11/07,11/03 |
Fitting ideals, Projective modules, Flat modules, Injective modules. |
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第9週 |
11/14,11/10 |
Graded modules, Extension of scalars, Algebras, Graded algebras, Symmetric algebras, Exterior algebras. |
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第10週 |
11/21,11/17 |
Nakayama lemma, Transcendence basis, Noether's normalization lemma. |
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第11週 |
11/28,11/24 |
Burnsides theorem on solvable groups, character relations. Invertible ideals. |
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第12週 |
12/05,12/01 |
Schur's lemma, Kummer theory, Spec(R). |
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第13週 |
12/12,12/08 |
Primary decomposition, Non-commutative rings, Quaternion algebras, Division algebras. |
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第14週 |
12/19,12/15 |
Artin-Rees lemma. Completions. |
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第15週 |
12/26,12/22 |
Hilbert polynomials, Valuations. |
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第16週 |
1/02,12/29 |
Krull intersection theorem, Krull's principal ideal theorem. |
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第17週 |
1/09,1/05 |
Snake lemma, rank of projective modules. |
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